Question

(6) Determine the truth value of the statement ∃x∀y(x lessthanorequalto y2) if the domain for the
variables consists of
(a) The positive real numbers.
(b) The integers.
(c) The nonzero real numbers.

Answer 1

We have the predicate $P(x,y)="x\leq y^2"$ (two variables). We have (a) $\R^+$ as a domain for both of the variables. We should determine for which values of $x$ and $y$ the predicate $\exist x\forall y (x\leq y^2)=1.$

Consider the predicate $S(x)=\forall y (x\leq y^2).$

Here $y$ should be greater or equal to $\sqrt{x}$. $\text{Then } S(x)=1, \exist x S (x)=1.\\$

So for all positive real numbers $x$ and $y\geq \sqrt{x}$ the predicate $\exist x\forall y (x\leq y^2)=1.$

(b) $\Z$

If $x\geq 0$ then $y\geq \sqrt{x}.$

If $x<0$ then $\forall y.$

(c) $\R-\{0\}$

If $x>0$ then $y\geq \sqrt{x}.$

If $x<0$ then $\forall y.$